GMAT Math : Algebra

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Example Questions

Example Question #62 : Functions/Series

What is the sum of the even terms from 2 to 60?

Possible Answers:

465

900

600

630

930

Correct answer:

930

Explanation:

We should notice that since we have a sequence of even numbers, we can factor out, so we can rewrite it as :

2+4+6+...+60=2(1+2+3+... + 30)

We can calculate the summation of all numbers from 1 to with the formula $\frac{n^{2}+n}{2}$ ; so, we simply have to plug in 30 for and multiply this formula by two:

$2(\frac{30^{n}+30}{2})= 930$

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Example Question #211 : Algebra

What is the sum of the sequence of all terms from 120 to 160 inclusive?

Possible Answers:

7140

5740

5840

2870

12880

Correct answer:

5740

Explanation:

The formula for the summation of consecutive terms is $\frac{n^{2}+n}{2}$ , which gives the sum of all terms from 0 to . We can apply the formula to get the summation of all consecutive terms from 1 to 160. To figure out the summation starting from 120, we simply have to subtract the summation of all terms from 1 to 119. (We don't want to include 120 since we want it in our summation.)

$\frac{160^{2}+160}{2}-\frac{119^{2}+119}{2} = 12880-7140= 5740$

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Example Question #1291 : Problem Solving Questions

What is the sum of all odd numbers from 0 to 59 inclusive?

Possible Answers:

731

870

900

873

360

Correct answer:

900

Explanation:

We can manipulate summations to make them easier to work with. Here, we are asked for the sum of odd terms from 1 to 59. We can calculate this by subtracting the summation of the even terms from the summation of all numbers from 1 to 59, using the formula $\frac{n^{2}+n}{2}$ to sum all terms from 1 to .

In other words, we have to calculate $\frac{59^{2}+59}{2}-\frac{2\cdot (29^{2}+29)}{2}$ , since the even numbers are given by 2+4+6...+58 or 2(1+2+3...+29) .

We obtain the final answer 900.

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Example Question #212 : Algebra

Define f (x) = x - |x | . Which of the following would be a valid alternative way of expressing the definition of ?

Possible Answers:

$f (x) = \left\{\begin{matrix} 0 & \textrm{ if } x < 0\\ 2x & \textrm{ if } x \geq 0 \end{matrix}\right.$

$f (x) = \left\{\begin{matrix} x & \textrm{ if } x < 0\\ 0 & \textrm{ if } x \geq 0 \end{matrix}\right.$

$f (x) = \left\{\begin{matrix} 1 & \textrm{ if } x < 1\\ 2x & \textrm{ if } x \geq 1 \end{matrix}\right.$

$f (x) = \left\{\begin{matrix} 1 & \textrm{ if } x < 1\\ x & \textrm{ if } x \geq 1 \end{matrix}\right.$

$f (x) = \left\{\begin{matrix} 2x & \textrm{ if } x < 0\\ 0 & \textrm{ if } x \geq 0 \end{matrix}\right.$

Correct answer:

$f (x) = \left\{\begin{matrix} 2x & \textrm{ if } x < 0\\ 0 & \textrm{ if } x \geq 0 \end{matrix}\right.$

Explanation:

By definition:

If $x \geq 0$ , then |x| = x ,and subsequently, f (x) = x - x = 0

If x < 0 , then |x| = -x ,and subsequently, $f (x) = x- \left ( - x \right ) = 2x$

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Example Question #73 : Understanding Functions

Piecewise

Let be the piecewise-defined function graphed above. Define the function $g (x) = x^{2} - 3$ .

Evaluate $\left (g \circ f \right ) (1)$ .

Possible Answers:

Correct answer:

Explanation:

$\left (g \circ f \right ) (1) = g [f(1)]$

As seen in the diagram below, the graph of includes the point (1, -9) .

Piecewise 1

Therefore, f(1) = -9 , and

$\left (g \circ f \right ) (1) = g (-9)$ .

$g (x) = x^{2} - 3$ , so

$g (-9) = (-9)^{2} - 3 = 81 - 3 = 78$ .

Therefore, $\left (g \circ f \right ) (1) = 78$ , the correct choice.

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Example Question #74 : Understanding Functions

Piecewise

Let be the piecewise-defined function graphed above. Define the function $g (x) = x^{2} - 3$ .

Evaluate $\left (f \circ g \right ) (1)$ .

Possible Answers:

Correct answer:

Explanation:

$\left (f \circ g \right ) (1) = f [g(1)]$

$g (x) = x^{2} - 3$ , so

$g (1) = 1^{2} - 3 = 1-3 = -2$ , so

$\left (f \circ g \right ) (1) = f (-2)$

As seen in the diagram below, the graph of includes the point (-2, -1) .

Piecewise 1

Therefore, f(-2) = -1 , and $\left (f \circ g \right ) (1) = -1$ , the correct choice.

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Example Question #75 : Understanding Functions

Piecewise

Let be the piecewise-defined function graphed above. Define function $g(x) = x^{2} - 11$ .

Evaluate $\left (g \circ f^{-1} \right ) (4)$ .

Possible Answers:

$45\frac{1}{2}$

$-10\frac{3}{4}$

4 is not in the domain of $g \circ f^{-1}$

Correct answer:

$-10\frac{3}{4}$

Explanation:

$\left (g \circ f^{-1} \right ) (4) = g [f ^{-1} (4)]$

$f ^{-1} (4) = c$ such that f(c)= 4 .As seen in the diagram below, the graph of includes the point $\left (\frac{1}{2}, 4 \right )$ , so $f ^{-1} (4) = \frac{1}{2}$ .

Piecewise 1

$\left (g \circ f^{-1} \right ) (4) = g \left ( \frac{1}{2} \right )$

$g(x) = x^{2} - 11$ , so

$g\left ( \frac{1}{2} \right )=\left ( \frac{1}{2} \right )^{2} - 11 = \frac{1}{4} -11 = -10 \frac{3}{4}$

$\left (g \circ f^{-1} \right ) (4) =-10 \frac{3}{4}$ , the correct choice.

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Example Question #76 : Understanding Functions

Piecewise

Let be the piecewise-defined function graphed above. Define function $g(x) = x^{2} - 11$ .

Evaluate $\left (f^{-1} \circ g \right ) (4)$ .

Possible Answers:

$-10\frac{3}{4}$

$45\frac{1}{2}$

4 is not in the domain of $f^{-1} \circ g$ .

Correct answer:

4 is not in the domain of $f^{-1} \circ g$ .

Explanation:

$\left (f^{-1} \circ g \right ) (4) = f ^{-1} [g(4)]$

$g(x) = x^{2} - 11$ , so

$g(4) = 4^{2} - 11 = 16 -11 = 5$

Therefore,

$\left (f^{-1} \circ g \right ) (4) = f ^{-1} (5)$ , which is equal to such that f(c) = 5 .

However, the range of the function , as can be seen from the diagram, is $[-9, -6] \cup [-4, 5)$ - 5 lies outside the range of , and, consequently, outside the domain of $f ^{-1}$ . Therefore, the expression $f ^{-1} (5)$ is undefined, and, equivalently, 4 is not in the domain of $f^{-1} \circ g$ .

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Example Question #77 : Understanding Functions

Piecewise

Let be the piecewise-defined function graphed above. Define a function g(x) = x-4 .

Evaluate $\left ( g^{-1} \circ f \right ) (6)$ .

Possible Answers:

$-10\frac{1}{2}$

6 is not in the domain of $g^{-1} \circ f$ .

$-2\frac{1}{2}$

$-6\frac{1}{2}$

$-8\frac{1}{2}$

Correct answer:

$-2\frac{1}{2}$

Explanation:

$\left ( g^{-1} \circ f \right ) (6) = g^{-1} [f(6)]$

From the diagram below, it can be seen that the point $\left ( 6, -6 \frac{1}{2}\right )$ is on the graph of .

Therefore, $f(6) = -6\frac{1}{2}$ , and

$\left ( g^{-1} \circ f \right ) (6) = g^{-1}\left ( -6 \frac{1}{2}\right )$

$g^{-1}\left ( -6 \frac{1}{2}\right ) = c$ if $g(c) =- 6 \frac{1}{2}$ , so, since

g(x) = x-4

$x-4 = -6 \frac{1}{2}$

$x = -2\frac{1}{2}$ ,

and $g^{-1}\left ( -6 \frac{1}{2}\right ) = -2\frac{1}{2}$

Therefore, $\left ( g^{-1} \circ f \right ) (6) = -2\frac{1}{2}$ .

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Example Question #78 : Understanding Functions

Piecewise

Let be the piecewise-defined function graphed above. Define a function g(x) = x-4 .

Evaluate $\left (f \circ g^{-1} \right ) (6)$ .

Possible Answers:

$\textup{6 is not in the domain of } f \circ g^{-1}$ .

$-8\frac{1}{2}$

$-6\frac{1}{2}$

$-2\frac{1}{2}$

$-10\frac{1}{2}$

Correct answer:

$\textup{6 is not in the domain of } f \circ g^{-1}$ .

Explanation:

$\left (f \circ g^{-1} \right ) (6) = f [g^{-1}(6)]$

$g^{-1}(6) = c$ if g(c) = 6 , so, since

g(x) = x-4

x-4 = 6

x = 10

Therefore, $g^{-1}(6) = 10$ , and

$\left (f \circ g^{-1} \right ) (6) = f (10)$

As can be seen from the diagram, however, the domain of is [-8, 7] . 10 is not in the domain of . Therefore, is not in the domain of $f \circ g^{-1}$ .

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #62 : Functions/Series

Example Question #211 : Algebra

Example Question #1291 : Problem Solving Questions

Example Question #212 : Algebra

Example Question #73 : Understanding Functions

Example Question #74 : Understanding Functions

Example Question #75 : Understanding Functions

Example Question #76 : Understanding Functions

Example Question #77 : Understanding Functions

Example Question #78 : Understanding Functions

All GMAT Math Resources

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